Differential Equations I Overview

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Questions and Answers

Which of the following functions is a general solution to the differential equation $y'' + y = 0$?

$c_1 ext{e}^{3x} + c_2 ext{e}^{-3x}$

$3 ext{e}^{3x}$

$c_1 ext{cos}x + c_2 ext{sin}x$ (correct)

$2 ext{e}^x + x ext{e}^x$

The function $u = e^{3x}$ is a solution to the differential equation $u'' + 2u' - 15u = 0$.

True (A)

What must be verified to establish that $y = 2e^x + xe^x$ is a particular solution of the second-order ODE $y'' - 2y' + y = 0$?

Substituting the function and its derivatives into the differential equation and checking against the initial conditions.

The general form for solving a second-order ODE with constant coefficients often results in sums of __________ functions.

exponential Signup and view all the answers

Match the following functions with their respective differential equations:

$ϕ = c_1 ext{cos}x + c_2 ext{sin}x$ = $y'' + y = 0$ $u = e^{3x}$ = $u'' + 2u' - 15u = 0$ $y = 2e^x + xe^x$ = $y'' - 2y' + y = 0$ Signup and view all the answers

Which of the following is an integrating factor for the equation $v' - 2v = x$?

$e^{-2x}$ (A) Signup and view all the answers

The general solution of the equation $y' + 4y = 0$ is $y = Ce^{-4x}$.

True (A) Signup and view all the answers

What is the first step in solving a first-order linear differential equation using the integrating factor method?

Find the integrating factor. Signup and view all the answers

In the example $u' - \frac{1}{x} u = x \cos x$, the integrating factor simplifies to $e^{-2\ln x} = \text{______}$.

x^{-2} Signup and view all the answers

To verify that $v = x - \frac{1}{2} + Ce^{-2x}$ is a solution to the equation $v' - 2v = x$, which operation must you perform?

Take the derivative of $v$ and substitute back into the original equation. (A) Signup and view all the answers

Particular solutions can be found by applying initial conditions to the general solution.

True (A) Signup and view all the answers

What is the result of applying the product rule to $e^{-2x} v'$ in the context of the differential equation?

e^{-2x} v = xe^{-2x} Signup and view all the answers

Match the function to its corresponding linear ordinary differential equation (ODE):

$y = Ce^{-4x}$ = $y' + 4y = 0$ $u = x^2 ext{sin}x + Cx^2$ = $u' - \frac{1}{x}u = x^2 ext{cos}x$ $v = x - \frac{1}{2} + Ce^{-2x}$ = $v' - 2v = x$ $C$ is a constant = Represents the general solution Signup and view all the answers

Which characteristic defines a homogeneous differential equation?

Set to zero (B) Signup and view all the answers

A particular solution to a differential equation has arbitrary constants.

False (B) Signup and view all the answers

What is the purpose of the Existence Theorem in differential equations?

To assure that a differential equation is solvable. Signup and view all the answers

A general solution of a differential equation has __ arbitrary constants.

n Signup and view all the answers

Match the following types of solutions to their definitions:

General Solution = Contains arbitrary constants and represents a family of solutions Particular Solution = Specific instance of a general solution with definite constants Homogeneous = No non-differential terms Non-Homogeneous = Contains non-differential terms Signup and view all the answers

Which of the following equations is considered non-homogeneous?

y'' + y' = 1 (D) Signup and view all the answers

A differential equation can have both a general solution and a particular solution.

True (A) Signup and view all the answers

What does IVP stand for in the context of differential equations?

Initial Value Problem Signup and view all the answers

Flashcards

Solution to y'' + y = 0

The function ϕ = c1 cos(x) + c2 sin(x), where c1 and c2 are arbitrary constants, satisfies the differential equation y'' + y = 0.

Verify u = e^(3x) solution

Substituting u = e^(3x) and its derivatives into u'' + 2u' - 15u = 0 results in zero, verifying it's a solution.